A. B. Quiriby on Crank Motion. ' 321 



the tendency which the power P'' has to produce rotation 

 when at S or when the crank is at a, is expressed by the pro- 

 duct of the tension of the shackle-bar, at that time, and the 

 perpendicular Cc, the distance between the centre of rota- 

 tion and the Une in which the value of the power is estimated: 

 wherefore the tendency of P" to produce rotation when the 



crank is at a, is properly expressed by ( P'' X -— ^ j X Cc 



=P"xCn : and in a similar manner it may be shewn that 

 the tendency of P" to produce rotation when the crank is 

 at d, is properly expressed by P'xC^. Hence the siim of 

 the tendencies of the power P" to produce rotation, when 

 the crank is at the points a and </, is expressed by P XCn 

 +P' yCv=P" X{Cn-\-Cv) — {because vt==tn)F' X2Ct. 



It now remains to compare this sum with the sum of the 

 tendencies which the equal power P has to produce rota- 

 tion, acting upon the crank (successively) at the two same 

 points, a and d. 



From what has been already stated, the sum of the ten- 

 dencies of P' to produce rotation, at the points a and d, is 

 expressed by P' Xam4-P'Xc?m=P' X2C<. 



Whence it appears (because P' and P" are equal) that 

 the sum of the tendencies of the power P" to produce rota- 

 tion, estimated at the two points a and d, is equal to the 

 sum of the tendencies which the equal power P' has to pro- 

 duce rotation, estimated at the same two points. And as 

 the same may be proved to be true at any two correspond- 

 ing points whatever, in the quadrants AD and D B, it fol- 

 lows that the tendency of P" to produce rotation, during 

 the time the crank is descending from A to B, is equal to 

 that which P' would have to produce rotation, descending 

 upon the crank from A to B, and acting at all times in a di- 

 rection parallel to SA. But it was proved that if P' de-= 

 scend upon the crank from A to B, and act at all times in 

 a direction parallel to SA, its effect during the time of de- 

 cent is =WxBA. 



Hence the effect of P ', during a descent of the crank from 

 A to B is also = W X BA. 



Wherefore in the cast in practice, as in the one before 

 demonstrated, there is no loss of the acting power. Q. E. D. 



A. B. QUINBY. 



Vol. Vn.— No. 2. 41 



